Studies on whether rent-a-reformatories are cheaper for taxpayers than government-run prisons have had conflicting results, largely because the data are hard to compare. Opinions also differ widely on whether private prisons, which tend to have lower guard-to-inmate ratios than public lockups, experience more violence. It's safe to say that if differences exist, they aren't very big.
Of course, private prisons are often established only where public prisons fail, or are on an unsustainable cost path. This will generally mean that when a private prison is built in a state, those public prisons that remain were already preforming better than the public prison that was replaced. Similarly, cross-state comparisons will suffer from the problem that states with private prisons were likely in a worse starting condition than states that have maintained public prisons only. Again, if the worst prisons in a state are replaced with privately-run prisons, then a zero difference between the new private prison and the remaining public prison is likely evidence of private success. This is because had the worst prisons remained public, they would (presumably) continue to do worse than the rest of the prisons in the state.